Question

An infinitely long current carrying wire carries current $i$. A charge of mass $m$ and charge $q$ is projected with speed $v$ parallel to the direction of current at a distance $r$ from it. Then, the radius of curvature at the point of projection is

• A. $\frac{2rmv}{q{\mu }_{0}i}$
• B. $\frac{2\pi rmv}{q{\mu }_{0}i}$
• C. $r$
• D. cannot be determined

Answer 1

Answer: (B)

$\frac{2\pi rmv}{q{\mu }_{0}i}$

Magnetic field due an infinitely long straight wire at a distance $r$ $B=\frac{{\mu }_{o}i}{2\pi r}$
Magnetic force acting on the charge $F=qvB$ $=\frac{{qv\mu }_{0}i}{2\pi r}$
This force makes the charge to trace a curved path of radius of curvature $R$.
Now $F_{centripetal}=F$

$\therefore$ $\frac{m{v}^2}{R}=\frac{{qv\mu }_{0}i}{2\pi r}$
$\Rightarrow R=\frac{2\pi rmv}{{q\mu }_{0}i}$